# (ln(3-x)-0.5lnx)/ln(x-1)=0.5

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First we want to simplify using `ln(a)-ln(b) = ln(a/b)` and we are also going to use `a*ln(c) = ln(c^a)` so

`(ln(3-x) - 1/2ln(x))/ln(x-1) = 1/2`

`(ln(3-x) - ln(x^(1/2))) = 1/2 ln(x-1)`

`ln((3-x)/x^(1/2)) = ln((x-1)^(1/2))` if ln(a) = ln(b) then a = b

`(3-x)/x^(1/2) = (x-1)^(1/2)` multiply both sides by `x^(1/2)` and `a^n b^b = (ab)^n`

`(3-x) = (x(x-1))^(1/2)` square both sides

`(3-x)^2 = x(x-1)` expanding both sides

`9 - 6x + x^2 = x^2 - x` subtracting `x^2` and adding `x` to both sides gives

`9 - 5x = 0` solving for x we get

`x = 9/5`

Now we need to check the answer

`(ln(3 - 9/5) - 1/2ln(9/5))/(ln(9/5 - 1)) = 1/2`

`(ln(15/5 - 9/5) - 1/2ln(9/5))/ln(9/5 - 5/5)) = 1/2`

`(ln(6/5) - ln(sqrt(9)/sqrt(5)))/ln(4/5) = 1/2`

`(ln(6/5) - ln(3/sqrt(5)))/ln(4/5) = 1/2`

`ln(6/5(sqrt(5)/3))/ln(4/5) = 1/2`

`ln((2sqrt(5))/5)/ln(4/5)=1/2`

`ln(sqrt(4*5)/sqrt(25))/ln(4/5) = 1/2`

`ln(sqrt(20/25))/ln(4/5) = 1/2 `

`ln(sqrt(4/5))/ln(4/5) = 1/2 `

`ln((4/5)^(1/2))/ln(4/5) = 1/2 `

`1/2(ln(4/5))/ln(4/5) = 1/2 `

`1/2 = 1/2` So we are correct

`x=9/5` is the solution.